Schur transform is a unitary operator that block diagonalizes the action of the symmetric and unitary groups on a tensor product V⊗n of a vector space V of dimension d with itself n times. In other words, Schur transform changes the basis from a computational to diagonal basis, which is a preferred basis for many problems. In mathematics, an unitary group of degree n, denoted U(n), is the group of n×n unitary matrices, with the group operation of matrix multiplication. In quantum mechanical computations, matrix diagonalization is one of the most frequently applied numerical processes. The basic reason is that the time-independent Schrödinger equation is an eigenvalue equation, albeit in most of the physical situations on an infinite dimensional space. A very common approximation is to truncate the infinite space to finite dimension, after which the Schrödinger equation can be formulated as an eigenvalue problem of a real symmetric, or complex Hermitian, matrix.
In a simplified way, Schur transform describes a unitary operation that converts a standard initial state to a “Schur basis,” from which (quantum) information can be easily extracted (measured). Generally, a set of quantum states, for example, in a high dimensional system include a substantial level of symmetry/redundancy, which are interchangeable without changing the high level system. Schur basis removes this symmetry/redundancy resulting in a more simplified system. Schur transform includes two group types associated with the symmetry/redundancy: an Unitary Group, and a Symmetric Group. Conventional methods use the Unitary Group to obtain the Schur basis. However, these methods become highly complex and impractical for high dimensional systems.
There are two versions of Schur transform, a weak transform and a strong transform. The week version performs a partial change of basis, whereas the strong version performs a full change of basis, depending on the problem and its requirements. Typically, there are three parameters involved with a Schur transform, d (qubit dimension), n (number of copies) and e (error). There exist weak transforms that are adequately efficient for all the three parameters. However, there is no strong transform that is efficient for all the three parameters.
Dave Bacon, Isaac L. Chuang and Aram W. Harrow, “Efficient quantum circuits for Schur and clebsch-gordan transforms,” Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '07, pages 1235-1244, Philadelphia, Pa., USA, 2007. Society for Industrial and Applied Mathematics. ISBN 978-0-898716-24-5 (“BCH”), the entire contents of which is hereby expressly incorporated by reference, using the Unitary Group, provided a quantum process for this transform that is polynomial in n, d and logε−1, where ε is the precision of the transform.